University of Kansas, Fall 2007
Philosophy 666: Rational Choice Theory
Ben Eggleston—eggleston@ku.edu
test on utility theory
(September 21, 2007)
Instructions:
- Answer all of the following questions on the answer sheets provided. You can write on this
list of
questions, but credit will be awarded only for answers written on answer
sheets.
- Do not access any book, notebook, newspaper, calculator, computer, cell
phone, or other possible source of inappropriate aid during the test, do not
leave the room before you are finished taking the test, and be sure to finish
the test within this 50-minute testing period—no credit will be given for any
work done after you access any possible source of inappropriate aid, after you
leave the room for any reason, or after the end of the testing period.
- When you are finished, be sure your name is written on each of your answer
sheets, and turn them in. You do not need to turn in this list of questions.
Questions:
- Consider the following preferences:
w P x
x P y
y P z
z P w
And consider the following proof (which may be entirely correct or may contain one or
more mistakes) purporting to show that these preferences violate the
transitivity condition:
# |
claim |
justification |
1 |
w P x |
given |
2 |
x P y |
given |
3 |
y P z |
given |
4 |
z P w |
given |
5 |
w P y |
1 and 2, transitivity condition |
6 |
w P z |
5 and 3, transitivity condition |
7 |
contradiction |
4 and 6, completeness condition |
What is the largest number n that makes the following sentence true?
“Lines 1 through n are all correct and correctly justified.” (If the
proof is entirely correct, give the number of the last line. If the proof has one or more
mistakes, give the number of the line immediately preceding the one containing
the first mistake.)
- Consider the following preferences. (They are numbered only for ease of
reference; the numbers do not have anything to do with utilities.)
1. b I a
2. i P h
3. h P c
4. a I c
5. g I h
6. c P i
7. d P b
8. f P d
If they
violate the transitivity condition, give an answer of the form ‘violation of
transitivity condition (n1, n2, n3,
. . .)’, where the numbers in parentheses refer to the line numbers of
preferences that form a cycle. List the line numbers in order so that, if the
preferences were written out, the option that come first alphabetically would
be mentioned first, and the option that is dispreferred in the first
preference is the one that is preferred in the second preference, and so on.
If the stated preferences violate the transitivity condition, this is all you
have to do, regardless of whether they satisfy or violate the completeness
condition. If the stated preferences satisfy the transitivity condition but
not the completeness condition, provide two or more (but as few as possible)
independent preference orderings that represent them. If the stated
preferences satisfy both the transitivity condition and the completeness
condition, provide a preference ordering that represents the stated
preferences.
- Suppose Miss Brooke has to take a trip, and she has to choose
between driving and flying. If she travels by car and there is an accident,
then her utility will be –50, and if she travels by car and there is no
accident, then her utility will be 10. If she travels by plane and there is an
accident (which, let us say, has a different probability than a car
accident), then her utility will be –800, and if she travels by plane and there
is no accident, then her utility will be 30. Set up and fill in a matrix for
this situation so that probabilities summing to 1 could be assigned to the
states of the world.
- Suppose an option has a utility of 9 in state S1, a
utility of 1 in
state S2, and a utility of 6 in state S3, with the probabilities of S1,
S2, and S3 being unspecified. Suppose also that, for
Mr. Lydgate, this option’s value according to the optimism/pessimism rule is the
same as its value according to the principle of maximizing expected utility
using the principle of insufficient reason. What is Mr. Lydgate’s optimism index?
- Suppose Mr. Casaubon prefers more money to less, but also prefers $1,000
to an option giving him a 1/4 chance at $4,100 and a 3/4 chance at $0.
Give utility assignments for the three dollar amounts that make the principle
of maximizing expected utility agree with Mr. Casaubon’s preferences.
- Suppose Mr. Tyke thinks Mr. Farebrother has a 50-percent chance at winning a
$10,000 prize for which Mr. Tyke is also competing. Mr. Tyke, thinking that he
is sure to win if Mr. Farebrother is out of the contest, offers Mr.
Farebrother $4,000 to
withdraw from the contest. If Mr. Farebrother accepts Mr. Tyke’s offer, is Mr.
Farebrother risk
averse, risk neutral, or risk seeking, or do we not have enough information to
say for sure? Explain your answer.
- If (1) Mr. Bulstrode’s insurance company sells an insurance policy to Mr.
Vincy,
(2) both parties have the same estimates of the probability and dollar amount
associated with the insured event, and (3) each party regards the transaction
as maximizing his or her or its expected utility, can we conclude that (4) Mr.
Vincy’s utility function for money (e.g., a function of the form u($x)
= f(x)) reflects greater risk aversion than is reflected by
Mr. Bulstrode’s
insurance company’s utility function for money? Why or why not?
For questions 8 and 9, assume that u($50) = u($40) + x, u($60) = u($50) + y,
and u($90) = u($60) + z, with x, y, and z being
positive numbers.
- Suppose Mr. Raffles prefers an option giving him a 1/3 chance at $90 and
a 2/3 chance at $40 to an option giving him a 1/3 chance at $50 and a 2/3
chance at $60. What constraint(s) on x, y, and z
(in addition to x > 0, y > 0, and z > 0) imply utility assignments for
the four dollar amounts that make the principle of maximizing expected utility
agree with these preferences? Any equation or inequality in your answer should
have z on the left and x and/or y on the right. Show your
work.
- Suppose Mr. Raffles prefers $60 to an option giving him a 1/2 chance at $90 and
a 1/2 chance at $50. What constraint(s) on x, y, and z
(in addition to x > 0, y > 0, and z > 0) imply utility assignments for
the four dollar amounts that make the principle of maximizing expected utility
agree with these preferences? Any equation or inequality in your answer should
have z on the left and x and/or y on the right. Show your
work.
- Is it possible for all of Mr. Raffles’s preferences to satisfy the conditions referred to in
the representation theorem? Why or why not?
Instructions, revisited:
As stated in item 3 of the instructions, you do not need to turn in this
list of questions.